Resistance-capacitance
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A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of
resistor A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active el ...
s and
capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...
s. It may be driven by a
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
or current source and these will produce different responses. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit. RC circuits can be used to filter a signal by blocking certain frequencies and passing others. The two most common RC filters are the high-pass filters and low-pass filters; band-pass filters and band-stop filters usually require
RLC filter An RLC circuit is an electrical circuit consisting of a resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the constituent componen ...
s, though crude ones can be made with RC filters.


Introduction

There are three basic, linear passive lumped analog circuit components: the resistor (R), the capacitor (C), and the
inductor An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a c ...
(L). These may be combined in the RC circuit, the RL circuit, the
LC circuit An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
, and the
RLC circuit An RLC circuit is an electrical circuit consisting of a electrical resistance, resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the ...
, with the acronyms indicating which components are used. These circuits, among them, exhibit a large number of important types of behaviour that are fundamental to much of analog electronics. In particular, they are able to act as passive filters. This article considers the RC circuit, in both series and
parallel Parallel is a geometric term of location which may refer to: Computing * Parallel algorithm * Parallel computing * Parallel metaheuristic * Parallel (software), a UNIX utility for running programs in parallel * Parallel Sysplex, a cluster of IBM ...
forms, as shown in the diagrams below.


Natural response

The simplest RC circuit consists of a resistor and a charged capacitor connected to one another in a single loop, without an external voltage source. Once the circuit is closed, the capacitor begins to discharge its stored energy through the resistor. The voltage across the capacitor, which is time-dependent, can be found by using Kirchhoff's current law. The current through the resistor must be equal in magnitude (but opposite in sign) to the time derivative of the accumulated charge on the capacitor. This results in the linear differential equation :C\frac + \frac=0 \,, where is the capacitance of the capacitor. Solving this equation for yields the formula for
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
: :V(t)=V_0 e^ \,, where is the capacitor voltage at time . The time required for the voltage to fall to is called the RC time constant and is given by, :\tau = RC \,. In this formula, is measured in seconds, in ohms and in farads.


Complex impedance

The complex impedance, (in
ohm Ohm (symbol Ω) is a unit of electrical resistance named after Georg Ohm. Ohm or OHM may also refer to: People * Georg Ohm (1789–1854), German physicist and namesake of the term ''ohm'' * Germán Ohm (born 1936), Mexican boxer * Jörg Ohm (b ...
s) of a capacitor with capacitance (in
farads The farad (symbol: F) is the unit of electrical capacitance, the ability of a body to store an electrical charge, in the International System of Units (SI). It is named after the English physicist Michael Faraday (1791–1867). In SI base units ...
) is :Z_C = \frac The complex frequency is, in general, a
complex number In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
, :s = \sigma + j \omega \,, where * represents the
imaginary unit The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
: , * is the
exponential decay A quantity is subject to exponential decay if it decreases at a rate proportional to its current value. Symbolically, this process can be expressed by the following differential equation, where is the quantity and (lambda) is a positive rate ...
constant (in
neper The neper (symbol: Np) is a logarithmic unit for ratios of measurements of physical field and power quantities, such as gain and loss of electronic signals. The unit's name is derived from the name of John Napier, the inventor of logarithms. As ...
s per second), and * is the
sinusoidal A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the '' sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in m ...
angular frequency In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
(in
radians per second The radian per second (symbol: rad⋅s−1 or rad/s) is the unit of angular velocity in the International System of Units (SI). The radian per second is also the SI unit of angular frequency, commonly denoted by the Greek letter ''ω'' (omega). ...
).


Sinusoidal steady state

Sinusoidal steady state is a special case in which the input voltage consists of a pure sinusoid (with no exponential decay). As a result, \sigma = 0 and the impedance becomes :Z_C = \frac = - \frac \,.


Series circuit

By viewing the circuit as a voltage divider, the
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge to m ...
across the capacitor is: :V_C(s) = \fracV_\mathrm(s) = \fracV_\mathrm(s) and the voltage across the resistor is: :V_R(s) = \fracV_\mathrm(s) = \fracV_\mathrm(s)\,.


Transfer functions

The transfer function from the input voltage to the voltage across the capacitor is :H_C(s) = \frac = \frac \,. Similarly, the transfer function from the input to the voltage across the resistor is :H_R(s) = \frac = \frac \,.


Poles and zeros

Both transfer functions have a single pole located at :s = -\frac \,. In addition, the transfer function for the voltage across the resistor has a
zero 0 (zero) is a number representing an empty quantity. In place-value notation Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...
located at the
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.


Gain and phase

The magnitude of the gains across the two components are :G_C = \big, H_C(j \omega) \big, = \left, \frac\ = \frac and :G_R = \big, H_R(j \omega) \big, = \left, \frac\ = \frac\,, and the phase angles are :\phi_C = \angle H_C(j \omega) = \tan^\left(-\omega RC\right) and :\phi_R = \angle H_R(j \omega) = \tan^\left(\frac\right)\,. These expressions together may be substituted into the usual expression for the
phasor In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
representing the output: :\begin V_C &= G_C V_\mathrm e^ \\ V_R &= G_R V_\mathrm e^\,. \end


Current

The current in the circuit is the same everywhere since the circuit is in series: :I(s) = \frac = \frac V_\mathrm(s)\,.


Impulse response

The
impulse response In signal processing and control theory, the impulse response, or impulse response function (IRF), of a dynamic system is its output when presented with a brief input signal, called an Dirac delta function, impulse (). More generally, an impulse ...
for each voltage is the inverse
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
of the corresponding transfer function. It represents the response of the circuit to an input voltage consisting of an impulse or
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...
. The impulse response for the capacitor voltage is :h_C(t) = \frac e^ u(t) = \frac e^ u(t)\,, where is the
Heaviside step function The Heaviside step function, or the unit step function, usually denoted by or (but sometimes , or ), is a step function, named after Oliver Heaviside (1850–1925), the value of which is zero for negative arguments and one for positive argume ...
and is the time constant. Similarly, the impulse response for the resistor voltage is :h_R(t) = \delta (t) - \frac e^ u(t) = \delta (t) - \frac e^ u(t)\,, where is the
Dirac delta function In mathematics, the Dirac delta distribution ( distribution), also known as the unit impulse, is a generalized function or distribution over the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire ...


Frequency-domain considerations

These are frequency domain expressions. Analysis of them will show which frequencies the circuits (or filters) pass and reject. This analysis rests on a consideration of what happens to these gains as the frequency becomes very large and very small. As : :G_C \to 0 \quad \mbox \quad G_R \to 1 \,. As : :G_C \to 1 \quad \mbox \quad G_R \to 0 \,. This shows that, if the output is taken across the capacitor, high frequencies are attenuated (shorted to ground) and low frequencies are passed. Thus, the circuit behaves as a '' low-pass filter''. If, though, the output is taken across the resistor, high frequencies are passed and low frequencies are attenuated (since the capacitor blocks the signal as its frequency approaches 0). In this configuration, the circuit behaves as a '' high-pass filter''. The range of frequencies that the filter passes is called its bandwidth. The point at which the filter attenuates the signal to half its unfiltered power is termed its cutoff frequency. This requires that the gain of the circuit be reduced to :G_C = G_R = \frac. Solving the above equation yields :\omega_\mathrm = \frac \quad \mbox \quad f_\mathrm = \frac which is the frequency that the filter will attenuate to half its original power. Clearly, the phases also depend on frequency, although this effect is less interesting generally than the gain variations. As : :\phi_C \to 0 \quad \mbox \quad \phi_R \to 90^ = \frac\mbox\,. As : :\phi_C \to -90^ = -\frac\mbox \quad \mbox \quad \phi_R \to 0\,. So at DC (0  Hz), the capacitor voltage is in phase with the signal voltage while the resistor voltage leads it by 90°. As frequency increases, the capacitor voltage comes to have a 90° lag relative to the signal and the resistor voltage comes to be in-phase with the signal.


Time-domain considerations

:''This section relies on knowledge of , the natural logarithmic constant''. The most straightforward way to derive the time domain behaviour is to use the
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
s of the expressions for and given above. This effectively transforms . Assuming a
step input Step(s) or STEP may refer to: Common meanings * Steps, making a staircase * Walking * Dance move * Military step, or march ** Marching Arts Films and television * ''Steps'' (TV series), Hong Kong * ''Step'' (film), US, 2017 Literature * '' ...
(i.e. before and then afterwards): :\begin V_\mathrm(s) &= V\cdot\frac \\ V_C(s) &= V\cdot\frac\cdot\frac \\ V_R(s) &= V\cdot\frac\cdot\frac \,. \end
Partial fraction In algebra, the partial fraction decomposition or partial fraction expansion of a rational fraction (that is, a fraction such that the numerator and the denominator are both polynomials) is an operation that consists of expressing the fraction a ...
s expansions and the inverse
Laplace transform In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform In mathematics, an integral transform maps a function from its original function space into another function space via integra ...
yield: :\begin V_C(t) &= V\left(1 - e^\right) \\ V_R(t) &= Ve^\,. \end These equations are for calculating the voltage across the capacitor and resistor respectively while the capacitor is charging; for discharging, the equations are vice versa. These equations can be rewritten in terms of charge and current using the relationships and (see
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
). Thus, the voltage across the capacitor tends towards as time passes, while the voltage across the resistor tends towards 0, as shown in the figures. This is in keeping with the intuitive point that the capacitor will be charging from the supply voltage as time passes, and will eventually be fully charged. These equations show that a series RC circuit has a time constant, usually denoted being the time it takes the voltage across the component to either rise (across the capacitor) or fall (across the resistor) to within of its final value. That is, is the time it takes to reach and to reach . The rate of change is a ''fractional'' per . Thus, in going from to , the voltage will have moved about 63.2% of the way from its level at toward its final value. So the capacitor will be charged to about 63.2% after , and essentially fully charged (99.3%) after about . When the voltage source is replaced with a short circuit, with the capacitor fully charged, the voltage across the capacitor drops exponentially with from towards 0. The capacitor will be discharged to about 36.8% after , and essentially fully discharged (0.7%) after about . Note that the current, , in the circuit behaves as the voltage across the resistor does, via
Ohm's Law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
. These results may also be derived by solving the
differential equation In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s describing the circuit: :\begin \frac &= C\frac \\ V_R &= V_\mathrm - V_C \,. \end The first equation is solved by using an
integrating factor In mathematics, an integrating factor is a function that is chosen to facilitate the solving of a given equation involving differentials. It is commonly used to solve ordinary differential equations, but is also used within multivariable calcul ...
and the second follows easily; the solutions are exactly the same as those obtained via Laplace transforms.


Integrator

Consider the output across the capacitor at ''high'' frequency, i.e. :\omega \gg \frac\,. This means that the capacitor has insufficient time to charge up and so its voltage is very small. Thus the input voltage approximately equals the voltage across the resistor. To see this, consider the expression for I given above: :I = \frac\,, but note that the frequency condition described means that :\omega C \gg \frac\,, so :I \approx \frac which is just
Ohm's Law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equat ...
. Now, :V_C = \frac\int_^I\,dt\,, so :V_C \approx \frac\int_^V_\mathrm\,dt\,, which is an
integrator An integrator in measurement and control applications is an element whose output signal is the time integral of its input signal. It accumulates the input quantity over a defined time to produce a representative output. Integration is an importan ...
''across the capacitor''.


Differentiator

Consider the output across the resistor at ''low'' frequency i.e., :\omega \ll \frac\,. This means that the capacitor has time to charge up until its voltage is almost equal to the source's voltage. Considering the expression for again, when :R \ll \frac\,, so :\begin I &\approx \frac\frac \\ V_\mathrm &\approx \frac = V_C \,.\end Now, :\begin V_R &= IR = C\fracR \\ V_R &\approx RC\frac\,, \end which is a
differentiator In electronics, a differentiator is a circuit that is designed such that the output of the circuit is approximately directly proportional to the rate of change (the time derivative) of the input. A true differentiator cannot be physically realized, ...
''across the resistor''. More accurate
integration Integration may refer to: Biology *Multisensory integration *Path integration * Pre-integration complex, viral genetic material used to insert a viral genome into a host genome *DNA integration, by means of site-specific recombinase technology, ...
and differentiation can be achieved by placing resistors and capacitors as appropriate on the input and
feedback Feedback occurs when outputs of a system are routed back as inputs as part of a chain of cause-and-effect that forms a circuit or loop. The system can then be said to ''feed back'' into itself. The notion of cause-and-effect has to be handled ...
loop of operational amplifiers (see ''
operational amplifier integrator An operational definition specifies concrete, replicable procedures designed to represent a construct. In the words of American psychologist S.S. Stevens (1935), "An operation is the performance which we execute in order to make known a concept." F ...
'' and '' operational amplifier differentiator'').


Parallel circuit

The parallel RC circuit is generally of less interest than the series circuit. This is largely because the output voltage is equal to the input voltage — as a result, this circuit does not act as a filter on the input signal unless fed by a current source. With complex impedances: :\begin I_R &= \frac \\ I_C &= j\omega C V_\mathrm\,. \end This shows that the capacitor current is 90° out of phase with the resistor (and source) current. Alternatively, the governing differential equations may be used: :\begin I_R &= \frac \\ I_C &= C\frac\,. \end When fed by a current source, the transfer function of a parallel RC circuit is: :\frac = \frac\,.


Synthesis

It is sometimes required to synthesise an RC circuit from a given
rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rat ...
in ''s''. For synthesis to be possible in passive elements, the function must be a
positive-real function Positive-real functions, often abbreviated to PR function or PRF, are a kind of mathematical function that first arose in electrical network synthesis. They are complex functions, ''Z''(''s''), of a complex variable, ''s''. A rational function is ...
. To synthesise as an RC circuit, all the critical frequencies (
poles and zeroes In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. In some sense, it is the simplest type of singularity. Technical ...
) must be on the negative real axis and alternate between poles and zeroes with an equal number of each. Further, the critical frequency nearest the origin must be a pole, assuming the rational function represents an impedance rather than an admittance. The synthesis can be achieved with a modification of the
Foster synthesis Network synthesis is a design technique for linear circuit, linear electrical circuits. Synthesis starts from a prescribed electrical impedance, impedance function of frequency or frequency response and then determines the possible networks that ...
or Cauer synthesis used to synthesise
LC circuit An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
s. In the case of Cauer synthesis, a ladder network of resistors and capacitors will result.Bakshi & Bakshi, pp. 3-30–3-37


See also

* RC time constant * RL circuit *
LC circuit An LC circuit, also called a resonant circuit, tank circuit, or tuned circuit, is an electric circuit consisting of an inductor, represented by the letter L, and a capacitor, represented by the letter C, connected together. The circuit can ac ...
*
RLC circuit An RLC circuit is an electrical circuit consisting of a electrical resistance, resistor (R), an inductor (L), and a capacitor (C), connected in series or in parallel. The name of the circuit is derived from the letters that are used to denote the ...
* Electrical network * List of electronics topics *
Step response The step response of a system in a given initial state consists of the time evolution of its outputs when its control inputs are Heaviside step functions. In electronic engineering and control theory, step response is the time behaviour of the out ...


References


Bibliography

* Bakshi, U.A.; Bakshi, A.V., ''Circuit Analysis - II'', Technical Publications, 2009 . * Horowitz, Paul; Hill, Winfield, ''The Art of Electronics'' (3rd edition), Cambridge University Press, 2015 {{ISBN, 0521809266. Analog circuits Electronic filter topology